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The porous medium equation (PME) is a typical nonlinear degenerate parabolicequation. An energetic variational approach has been studied in a recent work[6], in which the trajectory equation is obtained, and a few first orderaccurate numerical schemes have been developed and analyzed. In this paper, weconstruct and analyze a second order accurate numerical scheme in both time andspace. The unique solvability, energy stability are established, based on theconvexity analysis. In addition, we provide a detailed convergence analysis forthe proposed numerical scheme. A careful higher order asymptotic expansion isperformed and two step error estimates are undertaken. In more details, a roughestimate is needed to control the highly nonlinear term in a discrete$W^{1,\infty}$ norm, and a refined estimate is applied to derive the optimalerror order. Some numerical examples are presented as well.

A second order accurate numerical scheme for the porous medium equation by an energetic variational approach